The Mathematics of Financial Modelingand Investment Management

10-StochDiffEq Page 275 Wednesday, February 4, 2004 12:51 PM

Stochastic Differential Equations 275

and

φ( tx, )– φ( ty, ) +ψ( tx, )– ψ( ty, ) ≤ Dx ( – y) , t ∈ [ 0 , T] , x ∈ R

for appropriate constants C,D. The first condition is known as the lin-

ear growth condition, the last condition is the Lipschitz condition that

we encountered in ordinary differential equation (see Chapter 9). Sup-

pose that Z is a random variable independent of the σ -algebra ℑ ∞gener-

ated by B^2

t for t ≥ 0 such that EZ( ) < ∞. Then there is a unique

stochastic process X, defined for 0 ≤ t ≤ T, with time-continuous paths

such that X 0 = Z and such that the following equation is satisfied:

t t

Xt( ω , ) = X 0 + ∫ φ[ sX s , ( ω , )] sd + ∫ψ[ sX s , ( ω , )] dBs

0 0

The process X is called a strong solution of the above equation.

The above equation can be written in differential form as follows:

dX t ( ω , ) = φ[ tX t , ( ω , )] td + ψ[ tX t , ( ω , )] dBt

The differential form does not have an independent meaning; a differen-

tial stochastic equation is just a short albeit widely used way to write

the integral equation.

The key requirement of a strong solution is that the filtration ℑ t is

given and that the functions φ ,ψ are adapted to the filtration ℑ t. From

the economic (or physics) point of view, this requirement translates the

notion of causality. In simple terms, a strong solution is a functional of

the driving Brownian motion and of the “inputs” φ ,ψ. A strong solution

at time t is determined only by the “history” up to time t of the inputs

and of the random shocks embodied in the Brownian motion.

These conditions can be weakened. Suppose that we are given only

the two functions φ (t,x), ψ (t,x) and that we must construct a process Xt,

a Brownian motion Bt, and the relative filtration so that the above equa-

tion is satisfied. The equation still admits a unique solution with respect

to the filtration generated by the Brownian motion B. It is however only

a weak solution in the sense that, though there is no anticipation of

information, it is not a functional of a given Brownian motion.^3 Weak

and strong solutions do not necessarily coincide. However, any strong

solution is also a weak solution with respect to the same filtration.

(^3) See, for instance, Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Sto-
chastic Calculus (New York: Springer, 1991).